Chapter 21: Three-Digit by Two-Digit Division: Algorithms and Remainders
Learn Grade 5 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Three-Digit by Two-Digit Division: Algorithms and Remainders with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Division algorithms (a Grade 5 idea used in this chapter)
- Choosing quotient digits (a Grade 5 idea used in this chapter)
- Subtracting partial products (a Grade 5 idea used in this chapter)
- Bringing down place values (a Grade 5 idea used in this chapter)
- Remainders as leftovers (a Grade 5 idea used in this chapter)
- Remainders that round up a practical answer (a Grade 5 idea used in this chapter)
- Remainders written as fractions (a Grade 5 idea used in this chapter)
- Division word problems (a Grade 5 idea used in this chapter)
- Checking division using multiplication (a Grade 5 idea used in this chapter)
- Choosing a reasonable quotient (a Grade 5 idea used in this chapter)
How to Learn This Chapter
Read the explanation first, follow the examples one step at a time, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.
21.1 Division algorithms
Division algorithms develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Division algorithms?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Division algorithms develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Division algorithms develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Worked Example 2
Problem: What should you identify first before solving a problem about Division algorithms?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Division algorithms develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Division algorithms.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Division algorithms develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Division algorithms problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Division algorithms develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Division algorithms.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Division algorithms develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Division algorithms can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Tile a floor
- Write a simple rule: area = length × width.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Estimate tile quantity and include a small waste allowance.
Worked Example 7
Problem: Plan monthly savings
- Write a simple rule: total = starting amount + monthly deposit × months.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Predict when the savings goal is reached.
Worked Example 8
Problem: Simulate a die
- Write a simple rule: generate a random whole number from 1 to 6.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Repeat many times and compare frequencies.
Worked Example 9
Problem: Translate a point
- Write a simple rule: new x = x + 3; new y = y - 2.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Apply the same rule to every point.
Worked Example 10
Problem: Estimate paint needed
- Write a simple rule: paint = wall area ÷ coverage per can.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Round up because a partial can may not be enough.
Practice Exercise
Create one new question about Division algorithms. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
21.2 Choosing quotient digits
Choosing quotient digits is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Choosing quotient digits?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Choosing quotient digits is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Choosing quotient digits is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Choosing quotient digits?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Choosing quotient digits is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Choosing quotient digits.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Choosing quotient digits is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Choosing quotient digits problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Choosing quotient digits is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Choosing quotient digits.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Choosing quotient digits is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Choosing quotient digits can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Calculate 84 ÷ 7.
- Ask how many groups of 7 fit in 84.
- 7 × 12 = 84.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 12
Worked Example 7
Problem: Calculate 125 ÷ 8.
- Ask how many groups of 8 fit in 125.
- 8 × 15 = 120.
- Remainder = 5.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 15 R 5
Worked Example 8
Problem: Calculate 936 ÷ 9.
- Ask how many groups of 9 fit in 936.
- 9 × 104 = 936.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 104
Worked Example 9
Problem: Calculate 48 ÷ 12.
- Ask how many groups of 12 fit in 48.
- 12 × 4 = 48.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 4
Worked Example 10
Problem: Calculate 315 ÷ 5.
- Ask how many groups of 5 fit in 315.
- 5 × 63 = 315.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 63
Practice Exercise
Create one new question about Choosing quotient digits. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
21.3 Subtracting partial products
Subtracting partial products develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Subtracting partial products?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Subtracting partial products develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Subtracting partial products develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Worked Example 2
Problem: What should you identify first before solving a problem about Subtracting partial products?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Subtracting partial products develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Subtracting partial products.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Subtracting partial products develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Subtracting partial products problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Subtracting partial products develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Subtracting partial products.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Subtracting partial products develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Subtracting partial products can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Calculate 84 × 7.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 588
Worked Example 7
Problem: Calculate 125 × 8.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 1000
Worked Example 8
Problem: Calculate 936 × 9.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 8424
Worked Example 9
Problem: Calculate 48 × 12.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 576
Worked Example 10
Problem: Calculate 315 × 5.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 1575
Practice Exercise
Create one new question about Subtracting partial products. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
21.4 Bringing down place values
Bringing down place values helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Bringing down place values?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Bringing down place values helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Bringing down place values helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Worked Example 2
Problem: What should you identify first before solving a problem about Bringing down place values?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Bringing down place values helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Bringing down place values.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Bringing down place values helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Bringing down place values problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Bringing down place values helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Bringing down place values.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Bringing down place values helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Bringing down place values can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: What is the value of the first digit in 638,420,715?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 600,000,000
Worked Example 7
Problem: What is the value of the first digit in 92,305,004?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 90,000,000
Worked Example 8
Problem: What is the value of the first digit in 704,090,650?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 700,000,000
Worked Example 9
Problem: What is the value of the first digit in 18,765,432?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 10,000,000
Worked Example 10
Problem: What is the value of the first digit in 999,500,001?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 900,000,000
Practice Exercise
Create one new question about Bringing down place values. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
21.5 Remainders as leftovers
Remainders as leftovers is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Remainders as leftovers?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Remainders as leftovers is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Remainders as leftovers is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Remainders as leftovers?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Remainders as leftovers is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Remainders as leftovers.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Remainders as leftovers is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Remainders as leftovers problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Remainders as leftovers is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Remainders as leftovers.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Remainders as leftovers is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Remainders as leftovers can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Calculate 84 ÷ 7.
- Ask how many groups of 7 fit in 84.
- 7 × 12 = 84.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 12
Worked Example 7
Problem: Calculate 125 ÷ 8.
- Ask how many groups of 8 fit in 125.
- 8 × 15 = 120.
- Remainder = 5.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 15 R 5
Worked Example 8
Problem: Calculate 936 ÷ 9.
- Ask how many groups of 9 fit in 936.
- 9 × 104 = 936.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 104
Worked Example 9
Problem: Calculate 48 ÷ 12.
- Ask how many groups of 12 fit in 48.
- 12 × 4 = 48.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 4
Worked Example 10
Problem: Calculate 315 ÷ 5.
- Ask how many groups of 5 fit in 315.
- 5 × 63 = 315.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 63
Practice Exercise
Create one new question about Remainders as leftovers. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
21.6 Remainders that round up a practical answer
Remainders that round up a practical answer is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Remainders that round up a practical answer?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Remainders that round up a practical answer is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Remainders that round up a practical answer is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Remainders that round up a practical answer?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Remainders that round up a practical answer is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Remainders that round up a practical answer.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Remainders that round up a practical answer is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Remainders that round up a practical answer problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Remainders that round up a practical answer is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Remainders that round up a practical answer.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Remainders that round up a practical answer is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Remainders that round up a practical answer can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Calculate 84 ÷ 7.
- Ask how many groups of 7 fit in 84.
- 7 × 12 = 84.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 12
Worked Example 7
Problem: Calculate 125 ÷ 8.
- Ask how many groups of 8 fit in 125.
- 8 × 15 = 120.
- Remainder = 5.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 15 R 5
Worked Example 8
Problem: Calculate 936 ÷ 9.
- Ask how many groups of 9 fit in 936.
- 9 × 104 = 936.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 104
Worked Example 9
Problem: Calculate 48 ÷ 12.
- Ask how many groups of 12 fit in 48.
- 12 × 4 = 48.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 4
Worked Example 10
Problem: Calculate 315 ÷ 5.
- Ask how many groups of 5 fit in 315.
- 5 × 63 = 315.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 63
Practice Exercise
Create one new question about Remainders that round up a practical answer. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
21.7 Remainders written as fractions
Remainders written as fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Remainders written as fractions?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Remainders written as fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Remainders written as fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Remainders written as fractions?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Remainders written as fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Remainders written as fractions.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Remainders written as fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Remainders written as fractions problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Remainders written as fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Remainders written as fractions.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Remainders written as fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Remainders written as fractions can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Calculate 84 ÷ 7.
- Ask how many groups of 7 fit in 84.
- 7 × 12 = 84.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 12
Worked Example 7
Problem: Calculate 125 ÷ 8.
- Ask how many groups of 8 fit in 125.
- 8 × 15 = 120.
- Remainder = 5.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 15 R 5
Worked Example 8
Problem: Calculate 936 ÷ 9.
- Ask how many groups of 9 fit in 936.
- 9 × 104 = 936.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 104
Worked Example 9
Problem: Calculate 48 ÷ 12.
- Ask how many groups of 12 fit in 48.
- 12 × 4 = 48.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 4
Worked Example 10
Problem: Calculate 315 ÷ 5.
- Ask how many groups of 5 fit in 315.
- 5 × 63 = 315.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 63
Practice Exercise
Create one new question about Remainders written as fractions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
21.8 Division word problems
Division word problems develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Division word problems?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Division word problems develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Division word problems develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Worked Example 2
Problem: What should you identify first before solving a problem about Division word problems?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Division word problems develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Division word problems.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Division word problems develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Division word problems problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Division word problems develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Division word problems.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Division word problems develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Division word problems can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Division word problems in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Division word problems becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Division word problems and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Division word problems becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Division word problems using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Division word problems becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Division word problems problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Division word problems becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Division word problems could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Division word problems becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Division word problems. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
21.9 Checking division using multiplication
Checking division using multiplication develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Checking division using multiplication?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Checking division using multiplication develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Checking division using multiplication develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Worked Example 2
Problem: What should you identify first before solving a problem about Checking division using multiplication?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Checking division using multiplication develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Checking division using multiplication.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Checking division using multiplication develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Checking division using multiplication problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Checking division using multiplication develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Checking division using multiplication.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Checking division using multiplication develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Answer: Checking division using multiplication can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Find 10% of $90.
- Convert 10% to 0.1.
- Multiply by 90.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $9.00
Worked Example 7
Problem: Find 25% of $64.
- Convert 25% to 0.25.
- Multiply by 64.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $16.00
Worked Example 8
Problem: Find 15% of $140.
- Convert 15% to 0.15.
- Multiply by 140.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $21.00
Worked Example 9
Problem: Find 5% of $260.
- Convert 5% to 0.05.
- Multiply by 260.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $13.00
Worked Example 10
Problem: Find 20% of $75.
- Convert 20% to 0.2.
- Multiply by 75.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $15.00
Practice Exercise
Create one new question about Checking division using multiplication. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
21.10 Choosing a reasonable quotient
Choosing a reasonable quotient is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Choosing a reasonable quotient?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Choosing a reasonable quotient is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Choosing a reasonable quotient is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Choosing a reasonable quotient?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Choosing a reasonable quotient is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Choosing a reasonable quotient.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Choosing a reasonable quotient is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Choosing a reasonable quotient problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Choosing a reasonable quotient is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Choosing a reasonable quotient.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Choosing a reasonable quotient is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Choosing a reasonable quotient can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Calculate 84 ÷ 7.
- Ask how many groups of 7 fit in 84.
- 7 × 12 = 84.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 12
Worked Example 7
Problem: Calculate 125 ÷ 8.
- Ask how many groups of 8 fit in 125.
- 8 × 15 = 120.
- Remainder = 5.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 15 R 5
Worked Example 8
Problem: Calculate 936 ÷ 9.
- Ask how many groups of 9 fit in 936.
- 9 × 104 = 936.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 104
Worked Example 9
Problem: Calculate 48 ÷ 12.
- Ask how many groups of 12 fit in 48.
- 12 × 4 = 48.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 4
Worked Example 10
Problem: Calculate 315 ÷ 5.
- Ask how many groups of 5 fit in 315.
- 5 × 63 = 315.
- Remainder = 0.
Very beginner explanation: Division can be checked by multiplying the quotient by the divisor and adding any remainder.
Answer: 63
Practice Exercise
Create one new question about Choosing a reasonable quotient. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
Chapter Notes, Practice, and Common Mistakes
Chapter Notes
- Read the question before calculating.
- Represent the mathematics with a model, table, graph, diagram, number line, or equation when useful.
- Show the reasoning and check the final answer with a second method when possible.
Extra Practice
- Create and solve one original problem about Division algorithms.
- Create and solve one original problem about Choosing quotient digits.
- Create and solve one original problem about Subtracting partial products.
- Create and solve one original problem about Bringing down place values.
- Create and solve one original problem about Remainders as leftovers.
- Create and solve one original problem about Remainders that round up a practical answer.
Common Mistakes
- Skipping the meaning and trying to memorize a rule only.
- Using the wrong operation because the question was not read completely.
- Ignoring units, labels, place values, or the context of the problem.
- Not estimating or checking whether the final answer is reasonable.
30 Review Questions and Answers
Q1. What is the key idea in Division algorithms?
Answer: Division algorithms develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Q2. What is the key idea in Choosing quotient digits?
Answer: Choosing quotient digits is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q3. What is the key idea in Subtracting partial products?
Answer: Subtracting partial products develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Q4. What is the key idea in Bringing down place values?
Answer: Bringing down place values helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Q5. What is the key idea in Remainders as leftovers?
Answer: Remainders as leftovers is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q6. What is the key idea in Remainders that round up a practical answer?
Answer: Remainders that round up a practical answer is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q7. What is the key idea in Remainders written as fractions?
Answer: Remainders written as fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q8. What is the key idea in Division word problems?
Answer: Division word problems develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Q9. What is the key idea in Checking division using multiplication?
Answer: Checking division using multiplication develops accurate calculation and problem solving. Learners choose a strategy, show the steps, estimate when useful, and check the result with an inverse operation or another method.
Q10. What is the key idea in Choosing a reasonable quotient?
Answer: Choosing a reasonable quotient is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q11. Why should you estimate before or after a calculation?
Answer: Estimation gives a reasonable range and can reveal place-value or operation mistakes.
Q12. Why are labels and units important?
Answer: They show what a number represents and help prevent mixing unlike quantities.
Q13. How can a diagram help solve a problem?
Answer: A diagram makes quantities and relationships visible before calculation.
Q14. How can inverse operations check an answer?
Answer: An inverse operation reverses the original operation and should recover the starting value when the work is correct.
Q15. Why should you show steps?
Answer: Showing steps makes reasoning clear and helps find where an error happened.
Q16. What should you do after making a mistake?
Answer: Find the step where the reasoning changed, correct it, and try a similar problem again.
Q17. How can a number line support reasoning?
Answer: It shows order, distance, relative size, fractions, decimals, and inequality solutions visually.
Q18. When is a table useful?
Answer: A table organizes related values so patterns and comparisons are easier to see.
Q19. When is a graph useful?
Answer: A graph makes trends, comparisons, locations, or data patterns visible.
Q20. How do you decide which operation to use?
Answer: Use the meaning of the situation: combine, compare, find a difference, form equal groups, or find how many groups fit.
Q21. Why should you check place value?
Answer: A digit or decimal has a different value depending on its position.
Q22. What makes an answer reasonable?
Answer: It fits the context, has the expected size and units, and agrees with an estimate or second method.
Q23. How can you explain mathematical reasoning clearly?
Answer: Name the information, the rule or operation, the steps, and why the final answer fits the problem.
Q24. Why can more than one strategy be correct?
Answer: Different valid strategies can represent the same mathematical relationship and lead to the same result.
Q25. How should you approach a difficult Grade 5 problem?
Answer: Break it into smaller parts, solve one part at a time, and check each result before continuing.
Q26. How can you practise Three-Digit by Two-Digit Division: Algorithms and Remainders effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q27. How can you practise Three-Digit by Two-Digit Division: Algorithms and Remainders effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q28. How can you practise Three-Digit by Two-Digit Division: Algorithms and Remainders effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q29. How can you practise Three-Digit by Two-Digit Division: Algorithms and Remainders effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q30. How can you practise Three-Digit by Two-Digit Division: Algorithms and Remainders effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.